\chapter{Formule che nessuno ricorda mai}
\section{Trigonometria}
\[\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}\qquad a^2=b^2+c^2-2cb\cos\alpha\]
\[\frac{a+b}{a-b}=\frac{\tan\frac{1}{2}\left(\alpha+\beta\right)}{\tan\frac{1}{2}\left(\alpha-\beta\right)}\qquad\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}\]
\[\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}}\qquad\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}}\]
\[\sin\alpha\cos\beta=\frac{1}{2}\left[\sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)\right]\]
\[\cos\alpha\cos\beta=\frac{1}{2}\left[\cos\left(\alpha+\beta\right)+\cos\left(\alpha-\beta\right)\right]\]
\[\sin\alpha\sin\beta=\frac{1}{2}\left[\cos\left(\alpha-\beta\right)-\cos\left(\alpha+\beta\right)\right]\]
\[\sin\delta+\sin\gamma=2\sin\frac{\delta+\gamma}{2}\cos\frac{\delta-\gamma}{2}\]
\[\sin\delta-\sin\gamma=2\cos\frac{\delta+\gamma}{2}\sin\frac{\delta-\gamma}{2}\]
\[\cos\delta-\cos\gamma=2\cos\frac{\delta+\gamma}{2}\sin\frac{\delta-\gamma}{2}\]
\[\cos\delta+\cos\gamma=2\cos\frac{\delta+\gamma}{2}\cos\frac{\delta-\gamma}{2}\]
\[\cos\delta-\cos\gamma=2\sin\frac{\delta+\gamma}{2}\sin\frac{\delta-\gamma}{2}\]
\section{Sviluppi in serie di potenze}
\[e^x=\sum_{k=0}^\infty\frac{x^k}{k!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\cdots\]
\[\log(1+x)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{x^k}{k}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots\]
\[(1+x)^\alpha=\sum_{k=0}^\infty \binom{\alpha}{k}x^k=1+\binom{\alpha}{1}x+\binom{\alpha}{2}x^2+\binom{\alpha}{3}x^3+\binom{\alpha}{4}x^4+\cdots\]\footnote{\[\binom{\alpha}{y}=\frac{\overbrace{\alpha\left(\alpha-1\right)\cdots\left(\alpha-y+1\right)}^{\text{$y$ volte}}}{y!}\]}
\section{Operatori vettoriali}
\subsection{Gradiente}
\subsubsection{Coordinate Cartesiane}
\[\ve\nabla\varphi = \left(\frac{\partial\varphi}{\partial x},\frac{\partial\varphi}{\partial y},\frac{\partial\varphi}{\partial z}\right)\]
\subsubsection{Coordinate Cilindriche}
\[\left(\ve\nabla\varphi\right)_r=\frac{\partial\varphi}{\partial r}\qquad\left(\ve\nabla\varphi\right)_\phi=\frac{1}{r}\frac{\partial\varphi}{\partial\phi}\qquad\left(\ve\nabla\varphi\right)_w=\frac{\partial\varphi}{\partial w}\]
\subsubsection{Coordinate Sferiche}
\[\left(\ve\nabla\varphi\right)_r=\frac{\partial\varphi}{\partial r}\qquad\left(\ve\nabla\varphi\right)_\theta=\frac{\partial\varphi}{\partial\theta}\qquad\left(\ve\nabla\varphi\right)_\phi=\frac{1}{r\sin\theta}\frac{\partial\varphi}{\partial\phi}\]
\subsection{Divergenza}
\subsection{Rotore}
\subsection{Laplaciano}
\[\nabla^2\varphi=\ve\nabla\cdot\ve\nabla\varphi=\diver\grad\varphi\]
\[\nabla^2\ve F=\grad\diver\ve F-\rot\rot\ve F\]
\subsection{Relazioni}
\[\rot\grad\varphi=0\]
\[\diver\rot\ve F=0\]
\[\rot\rot\ve F=\grad\diver\ve F-\nabla^2\ve F\]